The Riemann sum approximates the net decrease in the volume of the balloon from t = 1 minute to t = 5 minutes. The product of the Riemann sum is in units of minutes. The Riemann sum approximates the net increase in the volume of the balloon from t = 1 minute to t = 5 minutes. The product of the Riemann sum is in units of liters. The product of the Riemann sum is in units of liters per minute.
The Riemann sum approximates the net decrease in the volume of the balloon from t = 1 minute to t = 5 minutes. The product of the Riemann sum is in units of minutes. The Riemann sum approximates the net increase in the volume of the balloon from t = 1 minute to t = 5 minutes. The product of the Riemann sum is in units of liters. The product of the Riemann sum is in units of liters per minute.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![The table shows the rate of change of volume \( V \) with respect to time \( t \) (in liters per minute) of a balloon for selected times \( t \), in minutes.
\[
\begin{array}{c|ccccc}
t & 1 & 2.5 & 3 & 4 & 5 \\
\hline
\frac{dV}{dt} & 7 & 6 & 6 & 5 & 4 \\
\end{array}
\]
In this table:
- The first row (\( t \)) represents the selected times in minutes.
- The second row (\( \frac{dV}{dt} \)) represents the rate of change of volume of the balloon in liters per minute at each of the corresponding times.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe636121f-6982-47b6-a42b-1ffa9dd55a08%2F520c2c0b-443b-4c4d-8f39-873888570d83%2Ff6j5fh5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The table shows the rate of change of volume \( V \) with respect to time \( t \) (in liters per minute) of a balloon for selected times \( t \), in minutes.
\[
\begin{array}{c|ccccc}
t & 1 & 2.5 & 3 & 4 & 5 \\
\hline
\frac{dV}{dt} & 7 & 6 & 6 & 5 & 4 \\
\end{array}
\]
In this table:
- The first row (\( t \)) represents the selected times in minutes.
- The second row (\( \frac{dV}{dt} \)) represents the rate of change of volume of the balloon in liters per minute at each of the corresponding times.
Transcribed Image Text:### Understanding Riemann Sums
When studying integrals, the concept of the Riemann sum becomes crucial as it provides a method to approximate the integral of a function. The Riemann sum essentially adds up the areas of a series of rectangles that approximate the area under a curve. It's a foundational concept in calculus.
### True Statements About the Riemann Sum
Consider the following statements about the Riemann sum, specifically in the context where it is applied to approximate changes in volume over a given time interval:
1. **The Riemann sum approximates the net decrease in the volume of the balloon from \( t = 1 \) minute to \( t = 5 \) minutes.**
- Check if this is true. If the function represents a rate of decrease, the Riemann sum could approximate the total volume decrease.
2. **The product of the Riemann sum is in units of minutes.**
- Consider whether the units of the quantities involved in the sum produce this unit result.
3. **The Riemann sum approximates the net increase in the volume of the balloon from \( t = 1 \) minute to \( t = 5 \) minutes.**
- Check if this is true. If the function represents a rate of increase, the Riemann sum could approximate the total volume increase.
4. **The product of the Riemann sum is in units of liters.**
- Analyze whether this statement is valid based on the units of the quantities being summed.
5. **The product of the Riemann sum is in units of liters per minute.**
- Determine if the units of the resulting product match this unit result.
### Analysis of the Statements
- **Riemann sum appropriately calculates changes (increase or decrease) in volume wherever the function applied represents a rate of volume change.**
- **Unit Analysis:**
- If the function being integrated has units of, say, liters per minute (rate of volume change) and time intervals are in minutes, the resulting Riemann sum would have units of liters (since liters/minute multiplied by minutes results in liters).
- Conversely, if the function being summed has units literally in minutes, that could mean the quantities are simply time steps.
By evaluating these statements, one can better understand the nature of the Riemann sum within various contexts and units, reinforcing crucial concepts in integral calculus.
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